On the Nash Equilibria of a Duel with Terminal Payoffs
Abstract
:1. Introduction
2. Game Description
- The game stays in state 11 ad infinitum (no player is ever killed);
- At some the game moves to a state (one or both players are killed). These are terminal states, i.e., as soon as they are reached, the game terminates.
3. Stationary Equilibria
- The transition to state 10 gives payoff and takes place with probability ( shot and hit ) multiplied by ( either shot and missed or did not shoot);
- The transition to state 01 gives payoff and takes place with probability ( shot and hit ) multiplied by ( either shot and missed or did not shoot);
- The transition to state 00 gives payoff and takes place with probability ( shot and hit ) multiplied by ( shot and hit );
- The transition to state 11 gives payoff (it is as if the game starts from the beginning) and takes place with probability ( either shot and missed or did not shoot) multiplied by ( either shot and missed or did not shoot).
4. Connection to the Iterated Prisoner’s Dilemma
- The IPD is a deterministic game, while involves randomness;
- In the IPD, each player receives a payoff in every turn and the total payoff is the discounted (by a discount factor ) sum of turn payoffs, while in , payoff is obtained only at the final turn and is undiscounted;
- The IPD will last an infinite number of turns, while may (depending on the p values and the strategy used) terminate in a finite number of turns (in fact, it may be the case that it will terminate in a finite number of terms with probability one).
5. Non-Stationary Equilibria
- They can continue shooting in all subsequent turns, in which case, so will ;
- They can revert to not shooting, in which case, in the next turn, they are in the same situation as at the start of the game.
- Consider first the case in which adopts the strategy of shooting in each turn. Then we have
- Next consider the case in which alternates between shooting and not shooting. Then their payoff will beThe above equation holds because the expected payoff is computed by summing the following possibilities. will certainly shoot and then:
- (a)
- With probability , will kill and hence, receive payoff ;
- (b)
- With probability , will miss (and receive zero payoff) and in the next turn will shoot and kill ; this combination has probability and gives to payoff ;
- (c)
- With probability , will miss and in the next turn will shoot and miss ; this combination has probability and returns the game to the original state, in which receives payoff .
Simplifying the above equation and solving we obtainFor an NE we must have and this will hold whenHowever, from our assumption (10), we haveHence .
6. Conclusions
- sequential play, in which a single player is allowed to shoot in each turn;
- random play, in which the player allowed to shoot in each turn is chosen randomly and equi-probably.
Funding
Data Availability Statement
Conflicts of Interest
1 | |
2 | In the sequel we use the standard game theoretic notation by which , . The same notation is used for players, actions etc. |
3 | Several parts of this paper require rather involved algebraic calculations. We have always performed these using the computer algebra system Maple and afterwards verified the results by hand. |
4 | We should clarify at this point that, despite the use of the terms “cooperation” and “cooperative”, the duel is not a cooperative game in Shapley’s sense [34]. In other words, it does not involve external enforcement of cooperative behavior. Instead, the duel is a non-cooperative game and “cooperation” is used in the same sense as in the Prisoner’s Dilemma literature; i.e., “cooperation” is understood as a spontaneous emergence of coordinated moves due to the players’ selfish behavior, rather than due to an explicit alliance mechanism. |
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Kehagias, A. On the Nash Equilibria of a Duel with Terminal Payoffs. Games 2023, 14, 62. https://doi.org/10.3390/g14050062
Kehagias A. On the Nash Equilibria of a Duel with Terminal Payoffs. Games. 2023; 14(5):62. https://doi.org/10.3390/g14050062
Chicago/Turabian StyleKehagias, Athanasios. 2023. "On the Nash Equilibria of a Duel with Terminal Payoffs" Games 14, no. 5: 62. https://doi.org/10.3390/g14050062
APA StyleKehagias, A. (2023). On the Nash Equilibria of a Duel with Terminal Payoffs. Games, 14(5), 62. https://doi.org/10.3390/g14050062